Table of Contents
 What is Regression Analysis?
 Terminologies related to Regression

Types of Regressions
 Linear Regression
 Polynomial Regression
 Logistic Regression
 Quantile Regression
 Ridge Regression
 Lasso Regression
 ElasticNet Regression
 Principal Component Regression
 Partial Least Square Regression
 Support Vector Regression
 Ordinal Regression
 Poisson Regression
 Negative Binomial Regression
 QuasiPoisson Regression
 Cox Regression
 How to choose the correct Regression Model?
What is Regression Analysis?
Lets take a simple example : Suppose your manager asked you to predict annual sales. There can be a hundred of factors (drivers) that affects sales. In this case, sales is your dependent variable. Factors affecting sales are independent variables. Regression analysis would help you to solve this problem.
In simple words, regression analysis is used to model the relationship between a dependent variable and one or more independent variables.
It helps us to answer the following questions 
 Which of the drivers have a significant impact on sales.
 Which is the most important driver of sales
 How do the drivers interact with each other
 What would be the annual sales next year.
Terminologies related to regression analysis
1. OutliersSuppose there is an observation in the dataset which is having a very high or very low value as compared to the other observations in the data, i.e. it does not belong to the population, such an observation is called an outlier. In simple words, it is extreme value. An outlier is a problem because many times it hampers the results we get.
2. Multicollinearity
When the independent variables are highly correlated to each other then the variables are said to be multicollinear. Many types of regression techniques assumes multicollinearity should not be present in the dataset. It is because it causes problems in ranking variables based on its importance. Or it makes job difficult in selecting the most important independent variable (factor).
3. Heteroscedasticity
When dependent variable's variability is not equal across values of an independent variable, it is called heteroscedasticity. Example  As one's income increases, the variability of food consumption will increase. A poorer person will spend a rather constant amount by always eating inexpensive food; a wealthier person may occasionally buy inexpensive food and at other times eat expensive meals. Those with higher incomes display a greater variability of food consumption.
4. Underfitting and Overfitting
When we use unnecessary explanatory variables it might lead to overfitting. Overfitting means that our algorithm works well on the training set but is unable to perform better on the test sets. It is also known as problem of high variance.
When our algorithm works so poorly that it is unable to fit even training set well then it is said to underfit the data. It is also known as problem of high bias.
In the following diagram we can see that fitting a linear regression (straight line in fig 1) would underfit the data i.e. it will lead to large errors even in the training set. Using a polynomial fit in fig 2 is balanced i.e. such a fit can work on the training and test sets well, while in fig 3 the fit will lead to low errors in training set but it will not work well on the test set.
Regression : Underfitting and Overfitting 
Types of Regression
Every regression technique has some assumptions attached to it which we need to meet before running analysis. These techniques differ in terms of type of dependent and independent variables and distribution.1. Linear Regression
It is the simplest form of regression. It is a technique in which the dependent variable is continuous in nature. The relationship between the dependent variable and independent variables is assumed to be linear in nature. We can observe that the given plot represents a somehow linear relationship between the mileage and displacement of cars. The green points are the actual observations while the black line fitted is the line of regressionRegression Analysis 
When you haveonly 1 independent variable
and 1 dependent variable, it is called simple linear regression.
When you havemore than 1 independent variable
and 1 dependent variable, it is called Multiple linear regression.
Multiple Regression Equation
Here 'y' is the dependent variable to be estimated, and X are the independent variables and ε is the error term. βi’s are the regression coefficients.
Assumptions of linear regression:
 There must be a linear relation between independent and dependent variables.
 There should not be any outliers present.
 No heteroscedasticity
 Sample observations should be independent.
 Error terms should be normally distributed with mean 0 and constant variance.
 Absence of multicollinearity and autocorrelation.
Interpretation of regression coefficients
Let us consider an example where the dependent variable is marks obtained by a student and explanatory variables are number of hours studied and no. of classes attended. Suppose on fitting linear regression we got the linear regression as:
Marks obtained = 5 + 2 (no. of hours studied) + 0.5(no. of classes attended)
 If no. of hours studied and no. of classes are 0 then the student will obtain 5 marks.
 Keeping no. of classes attended constant, if student studies for one hour more then he will score 2 more marks in the examination.
 Similarly keeping no. of hours studied constant, if student attends one more class then he will attain 0.5 marks more.
library(datasets)
model = lm(Fertility ~ .,data = swiss)
lm_coeff = model$coefficients
lm_coeff
summary(model)
The output we get is:
> lm_coeff
(Intercept) Agriculture Examination Education Catholic 66.9151817 0.1721140 0.2580082 0.8709401 0.1041153 Infant.Mortality 1.0770481 > summary(model) Call: lm(formula = Fertility ~ ., data = swiss) Residuals: Min 1Q Median 3Q Max 15.2743 5.2617 0.5032 4.1198 15.3213 Coefficients: Estimate Std. Error t value Pr(>t) (Intercept) 66.91518 10.70604 6.250 1.91e07 *** Agriculture 0.17211 0.07030 2.448 0.01873 * Examination 0.25801 0.25388 1.016 0.31546 Education 0.87094 0.18303 4.758 2.43e05 *** Catholic 0.10412 0.03526 2.953 0.00519 ** Infant.Mortality 1.07705 0.38172 2.822 0.00734 **  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 7.165 on 41 degrees of freedom Multiple Rsquared: 0.7067, Adjusted Rsquared: 0.671 Fstatistic: 19.76 on 5 and 41 DF, pvalue: 5.594e10
2. Polynomial Regression
In the figure given below, you can see the red curve fits the data better than the green curve. Hence in the situations where the relation between the dependent and independent variable seems to be nonlinear we can deploy Polynomial Regression Models.
In case of multiple variables say X1 and X2, we can create a third new feature (say X3) which is the product of X1 and X2 i.e.
Firstly we read the data using read.csv( ) and divide it into the dependent and independent variable
data = read.csv("poly.csv")
x = data$Area
y = data$Price
model1 = lm(y ~x)
model1$fit
model1$coeff
> model1$fit 1 2 3 4 5 6 7 8 9 10 169.0995 178.9081 188.7167 218.1424 223.0467 266.6949 291.7068 296.6111 316.2282 335.8454 > model1$coeff (Intercept) x 120.05663769 0.09808581
new_x = cbind(x,x^2)
new_x
x [1,] 500 250000 [2,] 600 360000 [3,] 700 490000 [4,] 1000 1000000 [5,] 1050 1102500 [6,] 1495 2235025 [7,] 1750 3062500 [8,] 1800 3240000 [9,] 2000 4000000 [10,] 2200 4840000Now we fit usual OLS to the new data:
model2 = lm(y~new_x)
model2$fit
model2$coeff
The fitted values and regression coefficients of polynomial regression are:
> model2$fit 1 2 3 4 5 6 7 8 9 10 122.5388 153.9997 182.6550 251.7872 260.8543 310.6514 314.1467 312.6928 299.8631 275.8110 > model2$coeff (Intercept) new_xx new_x 7.684980e+01 4.689175e01 1.402805e04
library(ggplot2)
ggplot(data = data) + geom_point(aes(x = Area,y = Price)) +
geom_line(aes(x = Area,y = model1$fit),color = "red") +
geom_line(aes(x = Area,y = model2$fit),color = "blue") +
theme(panel.background = element_blank())
3. Logistic Regression
 In linear regression, range of ‘y’ is real line but here it can take only 2 values. So ‘y’ is either 0 or 1 but X'B is continuous thus we can’t use usual linear regression in such a situation.
 Secondly, the error terms are not normally distributed.
 y follows binomial distribution and hence is not normal.
Examples
 HR Analytics: IT firms recruit large number of people, but one of the problems they encounter is after accepting the job offer many candidates do not join. So, this results in cost overruns because they have to repeat the entire process again. Now when you get an application, can you actually predict whether that applicant is likely to join the organization (Binary Outcome  Join / Not Join).
 Elections: Suppose that we are interested in the factors that influence whether a political candidate wins an election. The outcome (response) variable is binary (0/1); win or lose. The predictor variables of interest are the amount of money spent on the campaign and the amount of time spent campaigning negatively.
Predicting the category of dependent variable for a given vector X of independent variables
Through logistic regression we have
Interpreting the logistic regression coefficients (Concept of Odds Ratio)
We fit logistic regression with glm( ) function and we set family = "binomial"
model < glm(Lung.Cancer..Y.~Smoking..X.,data = data, family = "binomial")
#Predicted Probablities
model$fitted.values
1 2 3 4 5 6 7 8 9 0.4545455 0.4545455 0.6428571 0.6428571 0.4545455 0.4545455 0.4545455 0.4545455 0.6428571 10 11 12 13 14 15 16 17 18 0.6428571 0.4545455 0.4545455 0.6428571 0.6428571 0.6428571 0.4545455 0.6428571 0.6428571 19 20 21 22 23 24 25 0.6428571 0.4545455 0.6428571 0.6428571 0.4545455 0.6428571 0.6428571
data$prediction < model$fitted.values>0.5
> data$prediction [1] FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE TRUE TRUE [16] FALSE TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE
4. Quantile Regression
In linear regression, we predict the mean of the dependent variable for given independent variables. Since mean does not describe the whole distribution, so modeling the mean is not a full description of a relationship between dependent and independent variables. So we can use quantile regression which predicts a quantile (or percentile) for given independent variables.
The term “quantile” is the same as “percentile”
Basic Idea of Quantile Regression: In quantile regression we try to estimate the quantile of the dependent variable given the values of X's. Note that the dependent variable should be continuous.
The quantile regression model:
For qth quantile we have the following regression model:
This seems similar to linear regression model but here the objective function we consider to minimize is:
where q is the qth quantile.
If q = 0.5 i.e. if we are interested in the median then it becomes median regression (or least absolute deviation regression) and substituting the value of q = 0.5 in above equation we get the objective function as:
Interpreting the coefficients in quantile regression:
 Quite beneficial when heteroscedasticity is present in the data.
 Robust to outliers
 Distribution of dependent variable can be described via various quantiles.
 It is more useful than linear regression when the data is skewed.
Disclaimer on using quantile regression!
install.packages("quantreg")
library(quantreg)
model1 = rq(Fertility~.,data = swiss,tau = 0.25)
summary(model1)
tau: [1] 0.25 Coefficients: coefficients lower bd upper bd (Intercept) 76.63132 2.12518 93.99111 Agriculture 0.18242 0.44407 0.10603 Examination 0.53411 0.91580 0.63449 Education 0.82689 1.25865 0.50734 Catholic 0.06116 0.00420 0.22848 Infant.Mortality 0.69341 0.10562 2.36095
model2 = rq(Fertility~.,data = swiss,tau = 0.5)
summary(model2)
tau: [1] 0.5 Coefficients: coefficients lower bd upper bd (Intercept) 63.49087 38.04597 87.66320 Agriculture 0.20222 0.32091 0.05780 Examination 0.45678 1.04305 0.34613 Education 0.79138 1.25182 0.06436 Catholic 0.10385 0.01947 0.15534 Infant.Mortality 1.45550 0.87146 2.21101
model3 = rq(Fertility~.,data = swiss, tau = seq(0.05,0.95,by = 0.05))
quantplot = summary(model3)
quantplot
plot(quantplot)We get the following plot:
Various quantiles are depicted by X axis. The red central line denotes the estimates of OLS coefficients and the dotted red lines are the confidence intervals around those OLS coefficients for various quantiles. The black dotted line are the quantile regression estimates and the gray area is the confidence interval for them for various quantiles. We can see that for all the variable both the regression estimated coincide for most of the quantiles. Hence our use of quantile regression is not justifiable for such quantiles. In other words we want that both the red and the gray lines should overlap as less as possible to justify our use of quantile regression.
5. Ridge Regression
1. Regularization
Regularization is generally useful in the following situations:
 Large number of variables
 Low ratio of number observations to number of variables
 High MultiCollinearity
2. L1 Loss function or L1 Regularization
3. L2 Loss function or L2 Regularization
In general, L2 performs better than L1 regularization. L2 is efficient in terms of computation. There is one area where L1 is considered as a preferred option over L2. L1 has inbuilt feature selection for sparse feature spaces. For example, you are predicting whether a person is having a brain tumor using more than 20,000 genetic markers (features). It is known that the vast majority of genes have little or no effect on the presence or severity of most diseases.
Very Important Note:
We do not regularize the intercept term. The constraint is just on the sum of squares of regression coefficients of X's.
On solving the above objective function we can get the estimates of β as:
How can we choose the regularization parameter λ?
If we choose lambda = 0 then we get back to the usual OLS estimates. If lambda is chosen to be very large then it will lead to underfitting. Thus it is highly important to determine a desirable value of lambda. To tackle this issue, we plot the parameter estimates against different values of lambda and select the minimum value of λ after which the parameters tend to stabilize.
R code for Ridge Regression
X = swiss[,1]
y = swiss[,1]
library(glmnet)Using cv.glmnet( ) function we can do cross validation. By default alpha = 0 which means we are carrying out ridge regression. lambda is a sequence of various values of lambda which will be used for cross validation.
set.seed(123) #Setting the seed to get similar results.
model = cv.glmnet(as.matrix(X),y,alpha = 0,lambda = 10^seq(4,1,0.1))
best_lambda = model$lambda.min
ridge_coeff = predict(model,s = best_lambda,type = "coefficients")
ridge_coeff The coefficients obtained using ridge regression are:
6 x 1 sparse Matrix of class "dgCMatrix" 1 (Intercept) 64.92994664 Agriculture 0.13619967 Examination 0.31024840 Education 0.75679979 Catholic 0.08978917 Infant.Mortality 1.09527837
6. Lasso Regression
Note that lasso regression also needs standardization.
Advantage of lasso over ridge regression
Lasso regression can perform inbuilt variable selection as well as parameter shrinkage. While using ridge regression one may end up getting all the variables but with Shrinked Paramaters.
R code for Lasso Regression
#Creating dependent and independent variables.
X = swiss[,1]
y = swiss[,1]
#Setting the seed for reproducibility
set.seed(123)
model = cv.glmnet(as.matrix(X),y,alpha = 1,lambda = 10^seq(4,1,0.1))
#By default standardize = TRUE
#Taking the best lambda
best_lambda = model$lambda.min
lasso_coeff = predict(model,s = best_lambda,type = "coefficients")
lasso_coeff The lasso coefficients we got are:
6 x 1 sparse Matrix of class "dgCMatrix" 1 (Intercept) 65.46374579 Agriculture 0.14994107 Examination 0.24310141 Education 0.83632674 Catholic 0.09913931 Infant.Mortality 1.07238898
Which one is better  Ridge regression or Lasso regression?
It is a
combination of both L1 and L2 regularization
. R code for Elastic Net Regression
set.seed(123)The coeffients we obtained are:
model = cv.glmnet(as.matrix(X),y,alpha = 0.5,lambda = 10^seq(4,1,0.1))
#Taking the best lambda
best_lambda = model$lambda.min
en_coeff = predict(model,s = best_lambda,type = "coefficients")
en_coeff
6 x 1 sparse Matrix of class "dgCMatrix" 1 (Intercept) 65.9826227 Agriculture 0.1570948 Examination 0.2581747 Education 0.8400929 Catholic 0.0998702 Infant.Mortality 1.07757148. Principal Components Regression (PCR)
PCR is a regression technique which is widely used when you have many independent variables OR multicollinearity exist in your data. It is divided into 2 steps:
 Getting the Principal components
 Run regression analysis on principal components
 Dimensionality Reduction
 Removal of multicollinearity
Getting the Principal components
Principal components analysis is a statistical method to extract new features when the original features are highly correlated. We create new features with the help of original features such that the new features are uncorrelated.The first PC is having the maximum variance.
Similarly we can find the second PC U2 such that it is uncorrelated with U1 and has the second largest variance.
In a similar manner for 'p' features we can have a maximum of 'p' PCs such that all the PCs are uncorrelated with each other and the first PC has the maximum variance, then 2nd PC has the maximum variance and so on.
Drawbacks:
Principal Components Regression in R
data1 = longley[,colnames(longley) != "Year"]
View(data) This is how some of the observations in our dataset will look like:
We use pls package in order to run PCR.
install.packages("pls") library(pls)
pcr_model < pcr(Employed~., data = data1, scale = TRUE, validation = "CV")
summary(pcr_model)
Data: X dimension: 16 5 Y dimension: 16 1 Fit method: svdpc Number of components considered: 5 VALIDATION: RMSEP Crossvalidated using 10 random segments. (Intercept) 1 comps 2 comps 3 comps 4 comps 5 comps CV 3.627 1.194 1.118 0.5555 0.6514 0.5954 adjCV 3.627 1.186 1.111 0.5489 0.6381 0.5819 TRAINING: % variance explained 1 comps 2 comps 3 comps 4 comps 5 comps X 72.19 95.70 99.68 99.98 100.00 Employed 90.42 91.89 98.32 98.33 98.74
Here in the RMSEP the root mean square errors are being denoted. While in 'Training: %variance explained' the cumulative % of variance explained by principle components is being depicted. We can see that with 3 PCs more than 99% of variation can be attributed.
We can also create a plot depicting the mean squares error for the number of various PCs.
validationplot(pcr_model,val.type = "MSEP")By writing val.type = "R2" we can plot the R square for various no. of PCs.
validationplot(pcr_model,val.type = "R2")If we want to fit pcr for 3 principal components and hence get the predicted values we can write:
pred = predict(pcr_model,data1,ncomp = 3)
9. Partial Least Squares (PLS) Regression
It is an alternative technique of principal component regression when you have independent variables highly correlated. It is also useful when there are a large number of independent variables.
Difference between PLS and PCR
Both techniques create new independent variables called components which are linear combinations of the original predictor variables but PCR creates components to explain the observed variability in the predictor variables, without considering the response variable at all. While PLS takes the dependent variable into account, and therefore often leads to models that are able to fit the dependent variable with fewer components.PLS Regression in R
library(plsdepot)
data(vehicles)
pls.model = plsreg1(vehicles[, c(1:12,14:16)], vehicles[, 13], comps = 3)
# RSquare
pls.model$R2
10. Support Vector Regression
Support vector regression can solve both linear and nonlinear models. SVM uses nonlinear kernel functions (such as polynomial) to find the optimal solution for nonlinear models.
The main idea of SVR is to minimize error, individualizing the hyperplane which maximizes the margin.
library(e1071)
svr.model < svm(Y ~ X , data)
pred < predict(svr.model, data)
points(data$X, pred, col = "red", pch=4)
11. Ordinal Regression
Ordinal Regression is used to predict ranked values. In simple words, this type of regression is suitable when dependent variable is ordinal in nature. Example of ordinal variables  Survey responses (1 to 6 scale), patient reaction to drug dose (none, mild, severe).
Why we can't use linear regression when dealing with ordinal target variable?
In linear regression, the dependent variable assumes that changes in the level of the dependent variable are equivalent throughout the range of the variable. For example, the difference in weight between a person who is 100 kg and a person who is 120 kg is 20kg, which has the same meaning as the difference in weight between a person who is 150 kg and a person who is 170 kg. These relationships do not necessarily hold for ordinal variables.
library(ordinal)o.model < clm(rating ~ ., data = wine)
summary(o.model)
 Predicting the number of calls in customer care related to a particular product
 Estimating the number of emergency service calls during an event
 The dependent variable has a Poisson distribution.
 Counts cannot be negative.
 This method is not suitable on nonwhole numbers
pos.model<glm(breaks~wool*tension, data = warpbreaks, family=poisson)
summary(pos.model)
When the variance of count data is greater than the mean count, it is a case of overdispersion. The opposite of the previous statement is a case of underdispersion.
library(MASS)
nb.model < glm.nb(Days ~ Sex/(Age + Eth*Lrn), data = quine)
summary(nb.model)
qs.pos.model < glm(Days ~ Sex/(Age + Eth*Lrn), data = quine, family = "quasipoisson")
 Time from customer opened the account until attrition.
 Time after cancer treatment until death.
 Time from first heart attack to the second.
Logistic regression uses a binary dependent variable but ignores the timing of events.As well as estimating the time it takes to reach a certain event, survival analysis can also be used to compare timetoevent for multiple groups.
library(survival)
# Lung Cancer Data
# status: 2=death
lung$SurvObj < with(lung, Surv(time, status == 2))
cox.reg < coxph(SurvObj ~ age + sex + ph.karno + wt.loss, data = lung)
cox.reg
 If dependent variable is continuous and model is suffering from collinearity or there are a lot of independent variables, you can try PCR, PLS, ridge, lasso and elastic net regressions. You can select the final model based on Adjusted rsquare, RMSE, AIC and BIC.
 If you are working on count data, you should try poisson, quasipoisson and negative binomial regression.
 To avoid overfitting, we can use crossvalidation method to evaluate models used for prediction. We can also use ridge, lasso and elastic net regressions techniques to correct overfitting issue.
 Try support vector regression when you have nonlinear model.
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